S-Shaped Transition Curves as an Element of Reverse Curves in Road Design

Abstract

A road designing involves horizontal and vertical alignment. The horizontal geometry is formed by straight and curvilinear sections that are traditionally formed using circular and transition curves (mainly the clothoid). Different geometric systems that are designed using circular and transition curves are between others circular curves with symmetrical or unsymmetrical clothoids, combined curves, oval curves and reverse curves. Designing these systems is quite complex. Therefore, so-called S-shaped transition curves are an alternative to traditional approaches. These curves are known from literature and are modern geometric tools for the shaping of reverse curves. The paper analyses the basic geometric properties of these curves as well as compare to the geometry of the appropriate geometric systems, which are formed with clothoid or using S-shaped transition curves. In addition, a procedure for designing reverse curves using S-shaped transition curves was proposed. Another research topic was the comparison of the analysed reverse curves (created using polynomial transition curves) with traditional curves (created using the clothoid). The results of the studies, despite the noticeable differences in the geometry of the compared components, confirm the practical usefulness of the S-shaped transition curves for designing the geometry of the route.